A166962 Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.

1, 1, 1, 1, 12, 1, 1, 103, 69, 1, 1, 834, 2170, 316, 1, 1, 6685, 53910, 27830, 1329, 1, 1, 53496, 1219015, 1652300, 281195, 5412, 1, 1, 427987, 26455251, 81939195, 34800675, 2487917, 21781, 1, 1, 3423918, 563692024, 3700851816, 3327253410

Offset

1,5

Comments

Row sums are 1, 2, 14, 174, 3322, 89756, 3211420, 146132808, 8202467544, 554489060200,..

The recursion relation T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) connects several sequences for differing values of m. These are: m = 0 yields A008277, m = 1 yields A166960, m = 2 yields A166961, and m = 3 yields this sequence. These sequences are, in essence, a variation of Stirling numbers of the second kind. - G. C. Greubel, May 29 2016

Links

G. C. Greubel, Table of n, a(n) for the first 25 rows

Formula

T(n, k) = (3*n - 3*k + 1)*T(n - 1, k - 1) + k*(3*k - 2)*T(n - 1, k). - G. C. Greubel, May 29 2016

Example

1;
1, 1;
1, 12, 1;
1, 103, 69, 1;
1, 834, 2170, 316, 1;
1, 6685, 53910, 27830, 1329, 1;
1, 53496, 1219015, 1652300, 281195, 5412, 1;
1, 427987, 26455251, 81939195, 34800675, 2487917, 21781, 1;
1, 3423918, 563692024, 3700851816, 3327253410, 586846782, 20312292, 87300, 1;

Maple

A166962 := proc(n, k)
    if k = 1 or k = n then
        1;
    elif n <= 2 then
        1;
    else
        (3*n-3*k+1)*procname(n-1, k-1)+k*(3*k-2)*procname(n-1, k) ;
    end if;
end proc: # R. J. Mathar, Mar 26 2013

Mathematica

A[n_, 1] := 1; A[n_, n_] := 1; A[n_, k_] := (3*n - 3*k + 1)*A[n - 1, k - 1] + k*(3*k - 2)*A[n - 1, k]; Flatten[Table[A[n, k], {n, 10}, {k, n}]] (* modified by G. C. Greubel, May 29 2016 *)

Crossrefs

Cf. A008277, A166690, A166691.

Sequence in context: A155491 A142460 A156280 * A022175 A340427 A176627

Adjacent sequences: A166959 A166960 A166961 * A166963 A166964 A166965

Keyword

nonn,easy,tabl

Author

Roger L. Bagula and Mats Granvik, Oct 25 2009

Status

approved